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Trusted teacher: Digital suites courses I - General A numeric sequence is an application from N to R. • Bounded sequence A sequence (Un) is bounded if there exists a real A such that, for all n, Un ≤ A. We say that A is an upper bound of the series. A sequence (Un) is reduced if there exists a real number B such that, for all n, B ≤ one. One says that B is a lower bound of the sequence. A sequence is said to be bounded if it is both increased and reduced, that is to say if it exists M such that | Un | ≤ M for all n. • Convergent suite The sequence (Un) is convergent towards l ∈ R if: ∀ε> 0 ∃n0 ∈ N ∀n ≥ n0 | un − l | ≤ ε. A sequence which is not convergent is said to be divergent. When it exists, the limit of a sequence is unique. The deletion of a finite number of terms does not modify the nature of the sequence, nor its possible limit. Any convergent sequence is bounded. An unbounded sequence cannot therefore be convergent. • Infinite limits We say that the following (un) diverges Towards + ∞ if: ∀A> 0 ∃n0∈N ∀n ≥ n0 Un≥A Towards −∞ if: ∀A> 0 ∃n0∈N ∀n≤ n0 Un≤A. • Known limitations For k> 1, α> 0, β> 0 II Operations on suites • Algebraic operations If (un) and (vn) converge towards l and l ', then the sequences (un + vn), (λun) and (unvn) respectively converge towards l + l', ll and ll '. If (un) tends to 0 and if (vn) is bounded, then the sequence (unvn) tends to 0. • Order relation If (un) and (vn) are convergent sequences such that we have a ≤ vn for n≥n0, then we have: Attention, no analogous theorem for strict inequalities. • Framing theorem If, from a certain rank, un ≤xn≤ vn and if (un) and (vn) converge towards the same limit l, then the sequence (xn) is convergent towards l. III monotonous suites • Definitions The sequence (un) is increasing if un + 1≥un for all n; decreasing if un + 1≤un for all n; stationary if un + 1 = one for all n. • Convergence Any sequence of increasing and increasing reals converges. Any decreasing and underestimating sequence of reals converges. If a sequence is increasing and not bounded, it diverges towards + ∞. • Adjacent suites The sequences (un) and (vn) are adjacent if: (a) is increasing; (vn) is decreasing; If two sequences are adjacent, they converge and have the same limit. If (un) increasing, (vn) decreasing and un≤vn for all n, then they converge to l1 and l2. It remains to show that l1 = l2 so that they are adjacent. IV Extracted suites • Definition and properties - The sequence (vn) is said to be extracted from the sequence (un) if there exists a map φ of N in N, strictly increasing, such that vn = uφ (n). We also say that (vn) is a subsequence of (un). - If (un) converges to l, any subsequence also converges to l. If sequences extracted from (un) all converge to the same limit l, we can conclude that (un) converges to l if all un is a term of one of the extracted sequences studied. For example, if (u2n) and (u2n + 1) converge to l, then (un) converges to l. • Bolzano-Weierstrass theorem From any bounded sequence of reals, we can extract a convergent subsequence. V Suites de Cauchy • Definition A sequence (un) is Cauchy if, for any positive ε, there exists a natural integer n0 for which, whatever the integers p and q greater than or equal to n0, we have | up − uq | <ε. Be careful, p and q are not related. • Property A sequence of real numbers, or of complexes, converges if, and only if, it is Cauchy SPECIAL SUITES I Arithmetic and geometric sequences • Arithmetic sequences A sequence (un) is arithmetic of reason r if: ∀ n∈N un + 1 = un + r General term: un = u0 + nr. Sum of the first n terms: • Geometric sequences A sequence (un) is geometric of reason q ≠ 0 if: ∀ n∈N un + 1 = qun. General term: un = u0qn Sum of the first n terms: II Recurring suites • Linear recurrent sequences of order 2: - Such a sequence is determined by a relation of the type: (1) ∀ n∈N aUn + 2 + bUn + 1 + cUn = 0 with a ≠ 0 and c ≠ 0 and knowledge of the first two terms u0 and u1. The set of real sequences which satisfy the relation (1) is a vector space of dimension 2. We seek a basis by solving the characteristic equation: ar2 + br + c = 0 (E) - Complex cases a, b, c If ∆ ≠ 0, (E) has two distinct roots r1 and r2. Any sequence satisfying (1) is then like : where K1 and K2 are constants which we then express as a function of u0 and u1. If ∆ = 0, (E) has a double root r0 = (- b) / 2a. Any sequence satisfying (1) is then type: - Case a, b, c real If ∆> 0 or ∆ = 0, the form of the solutions is not modified. If ∆ <0, (E) has two conjugate complex roots r1 = α + iβ and r2 = α − iβ that we write in trigonometric form r1 = ρeiθ and r2 = ρe-iθ Any sequence satisfying (1) is then of the type: • Recurrent sequences un + 1 = f (un) - To study such a sequence, we first determine an interval I containing all the following values. - Possible limit If (un) converges to l and if f is continuous to l, then f (l) = l. - Increasing case f If f is increasing over I, then the sequence (un) is monotonic. The comparison of u0 and u1 makes it possible to know if it is increasing or decreasing. - Decreasing case f If f is decreasing over I, then the sequences (u2n) and (u2n + 1) are monotonic and of contrary Made by LEON
Math · Physics · Computer science
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Computer science
Trusted teacher: With over seven years of experience in teaching Computer Science & Information Technology (ICT), I have developed a strong expertise in delivering high-quality education across multiple internationally recognized curricula, including Cambridge IGCSE, GCSE, A-Levels, O-Levels, and Checkpoint. My passion lies in equipping students with coding, cybersecurity, and digital literacy skills, ensuring they are well-prepared for the evolving demands of the digital world. Expertise & Teaching Areas: ✅ Programming & Software Development: Python, Java, C++ ✅ Cybersecurity: Ethical hacking, data protection, network security ✅ Digital Literacy: ICT applications, online safety, cloud computing ✅ Data Science & AI: Data analysis, machine learning fundamentals ✅ Web Development: HTML, CSS, JavaScript Curriculum & Pedagogical Experience: 🔹 Cambridge IGCSE & GCSE ICT & Computer Science – Teaching core and extended syllabi, focusing on programming logic, databases, and networking. 🔹 Cambridge A-Levels & O-Levels Computer Science – Preparing students for advanced computing concepts, problem-solving, and algorithm development. 🔹 Cambridge Checkpoint ICT – Building foundational skills in digital technology and computer applications. Professional Impact: 📌 Mentored students to achieve top grades in Cambridge ICT & Computer Science exams. 📌 Developed interactive lesson plans integrating real-world applications of technology. 📌 Conducted coding boot camps and cybersecurity workshops to enhance practical learning. 📌 Guided students in project-based learning, including app development and website design. With a strong commitment to student-centered learning and technological innovation, I am dedicated to shaping future tech leaders and empowering learners with skills relevant to careers in technology, data science, and software development.
Computer science · Information technology
Trusted teacher: Python is a high level programming language. It's object oriented approach help programmers to write simple and logical codes. Due to its easy syntax, it is actually easier as compared to other programming languages. Having years of experience in coding language, I know how and where to start and move the topics at a good pace. I'll be starting with you from scratch and help you to build your concepts and master over the language. I am providing you a brief summary of our course which we generally follow but that doesn't mean that I am bound to just this much content. In case you have any doubt for any other specific topic related to the subject, you could always come up to me or you need any help in your assignments or projects, you are always welcome. Here is a brief overview of the topics we'll be covering : 1: Introduction To Python • Installation and Working with Python • Understanding Python variables • Python basic Operators • Understanding python blocks 2: Python Data Types • Declaring and using Numeric data types: int, float, complex • Using string data type and string operations • Defining list and list slicing • Use of Tuple data type 3: Python Program Flow Control • Conditional blocks using if, else and elif • Simple for loops in python • For loop using ranges, string, list and dictionaries • Use of while loops in python • Loop manipulation using pass, continue, break and else • Programming using Python conditional and loops block 4: Python Functions, Modules And Packages • Organizing python codes using functions • Organizing python projects into modules • Importing own module as well as external modules • Understanding Packages • Powerful Lamda function in python • Programming using functions, modules and external packages 5: Python String, List And Dictionary Manipulations • Building blocks of python programs • Understanding string in build methods • List manipulation using in build methods • Dictionary manipulation • Programming using string, list and dictionary in build functions 6: Python File Operation • Reading config files in python • Writing log files in python • Understanding read functions, read(), readline() and readlines() • Understanding write functions, write() and writelines() • Manipulating file pointer using seek • Programming using file operations 7: Python Object Oriented Programming – Oops • Concept of class, object and instances • Constructor, class attributes and destructors • Real time use of class in live projects • Inheritance , overlapping and overloading operators • Adding and retrieving dynamic attributes of classes • Programming using Oops support 8: Python Regular Expression • Powerful pattern matching and searching • Power of pattern searching using regex in python • Real time parsing of networking or system data using regex • Password, email, url validation using regular expression • Pattern finding programs using regular expression 9: Python Exception Handling • Avoiding code break using exception handling • Safe guarding file operation using exception handling • Handling and helping developer with error code • Programming using Exception handling 10: Python Database Interaction • SQL Database connection using python • Creating and searching tables • Reading and storing config information on database • Programming using database connections 11: Python Multithreading • Understanding threads • Forking threads • Synchronizing the threads • Programming using multithreading 12: Contacting User Through Emails Using Python • Installing smtp python module • Sending email • Reading from file and sending emails to all users addressing them directly for marketing 13: Python CGI Introduction • Writing python program for CGI applications • Creating menus and accessing files • Server client program In case you have any query, please feel free to contact me. Give yourself a chance for a better future. All the best
Computer science · Computer programming · Computer engineering
Trusted teacher: Are you preparing for the competitive entrance exams for the grandes écoles (X, Centrale, Mines, ENS, HEC, ESSEC, EDHEC, etc.)? Are you in a science prep course (MPSI, MP, PCSI, PC) or economics prep course (ECS, ECE, ECT) and are you looking to optimize your methodology, fill in your gaps, or aim for excellence? I am a certified mathematics teacher with 20 years of experience teaching preparatory classes and providing individual tutoring. I offer personalized private lessons in mathematics and physics, tailored to your level and goals. What I propose: In-depth preparation for written and oral competitive exams (central exams, joint banks, oral exams at X or HEC, etc.) Clear, structured courses, with course reminders, resolution methodology, tips and intensive practice on past papers Regular and personalized monitoring, identification of weak points, consolidation of acquired knowledge Training in rigorous writing, math/physics oral exams, time management and perspective-taking Materials provided: method sheets, past exam corrections, thematic summaries Concerned public : Students in MPSI, MP, PCSI, PC, ECS, ECE, ECT Students aiming for prestigious engineering schools or business schools Need for refresher training, intensive coaching, or targeted development? Why choose me? Recognized educational expertise Adaptation to each student profile Proven results: many students admitted to Polytechnique, CentraleSupélec, Mines, HEC, ESSEC, EDHEC... Courses at home or online (Zoom, Teams, etc.), depending on your preference. Contact me to discuss your needs and arrange an initial appointment.
Math · Physics · Computer science
Computer science · Microsoft excel
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Only reviews of students are published and they are guaranteed by Apprentus. Rated 4.8 out of 5 based on 66 reviews.

Python Step by step tutorial for beginners and intermediate (Amsterdam)
Izhar
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Izhar is an excellent tutor. He is very professional, knowledgeable, enlightening, insightful and adaptable/receptive to individual student needs. My tutoring session with him was really amazing and transformative for me and I highly recommend him. He keeps his lessons effective, engaging and fun. He customizes classes based upon your level of understanding. He conveys his knowledge of complex material in a manner which is easily understandable. If you want someone to work with you and ensure you progress like I have, go with Izhar.
Review by DANNY
Expert Math Tutor | School & University Level | Exam Prep & Confidence Boosting (Amsterdam)
Baia
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I couldn’t ask for a better tutor for my daughter! Baia is incredibly knowledgeable in math and algorithms, but what I truly think it sets her apart is her patience, kindness, and ability to make complex concepts easy to understand. She is always well-prepared and adapts her teaching style to fit my daughter’s needs, ensuring that learning is both effective and enjoyable. My daughter has gained so much confidence in her skills thanks to Baia’s guidance. I highly recommend her to anyone looking for an outstanding tutor!
Review by PATRICK REIS
Mastering Web Development: From Fundamentals to Advanced Techniques (Les Bons Villers)
Oussama
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I have no prior experience in this area and I took lessons with Ousamma. He has enough patience and a knack in guiding you step by step through the lessons and exercises in a very simple way to understand them well. He is an invested teacher that is kind and polite. He will help you to develop your abilities and to feel confident in order to achieve your goals. I warmly recommend him and extend my gratitude his great support throughout my learning journey.
Review by BEN
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